The Canonical Group of Transformations of a Gaussian or Hermitean Frame

We show that the lattice of any multidimensional Gaussian Gabor frame can be deformed using a Hamiltonian flow determined by the window, viewed as a stationary state of a certain Hamiltonian operator. Thus, to every such frame is intrinsically attached a canonical one-parameter group of symplectic transformations. We give applications to Gaussian and Germitean frames.